Groups Generated by reflections and aspherical manifolds not covered by Euclidean space
Groups Generated by reflections and aspherical manifolds not covered by Euclidean space
复制标题
DOI：
10.2307/2007079
复制
发表时间：
1983-03
影响因子：
4.9
通讯作者：
Michael W. Davis
中科院分区：
文献类型：
--
作者：
Michael W. Davis
一个考克斯特系统（r，V）是一个群r（一个“考克斯特群”）连同一组生成元V，使得V的每个元素都有阶2，并且使得r中的所有关系都是形式为（VW）m（v，w）= 1的关系的后果，其中v，w ∈ V和m（v，w）表示vw的阶。m（v，w）（它们是正整数或oo）显然决定了Coxeter系统直到同构。有限Coxeter群的列表很短且众所周知;然而，一般Coxeter群要灵活得多。事实上，集合V和m（v，w）可以被任意指定（至少如果V是有限的），仅受条件：m（v，v）= 1和m（v，w）= m（w，v）？2，如果v -# w。设（r，V）是Coxeter系统，X是Hausdorff空间，（Xv）v~v是V所标闭子空间的局部有限族（Xv称为X的“面元”）. (2))有一个经典的方法可以从这些数据中构造一个变换群。对于每个x E X，设V（x）表示V中的v的集合，使得x E Xv。对于V的每个子集S，设rs是由S生成的子群，设Xs是由下式定义的X的“面”：
A Coxeter system (r, V) is a group r (a "Coxeter group") together with a set of generators V such that each element of V has order two and such that all relations in r are consequences of relations of the form (VW)m(v, w) = 1, where v, w E V and m(v, w) denotes the order of vw. The m(v, w)'s (which are positive integers or oo) obviously determine the Coxeter system up to isomorphism. The list of finite Coxeter groups is short and well-known; however, general Coxeter groups are much more flexible. In fact, the set V and the m(v, w)'s can be specified arbitrarily (at least if V is finite) subject only to the conditions: m(v, v) = 1 and m(v, w) = m(w, v) ? 2 if v -# w. Suppose that (r, V) is a Coxeter system, that X is a Hausdorff space and that (Xv)v~v is a locally finite family of closed subspaces indexed by V. (The Xv are called the "panels" of X.(2)) There is a classical method for constructing a transformation group from these data. For each x E X, let V(x) denote the set of v in V such that x E Xv. For each subset S of V, let rs be the subgroup generated by S and let Xs be the "face" of X defined by